Penalty: finite element approximation of Stokes equations with slip boundary conditions

Abstract

We consider the finite element approximation of the stationary Stokes equations with slip boundary conditions on a domain with a smooth curved boundary. The slip boundary condition is imposed weakly with the penalty method on polygonal domains approaching the smooth domain. For Taylor-Hood elements, we derive error estimates which depend on the penalty parameter $$\varepsilon $$ ? , the disctretization parameter $$h$$ h and the approximation error of the normal to the boundary. In particular, if in the penalty term we use the normal to the polygonal boundary, the best convergence order is $$2/3$$ 2 / 3 and it is obtained with $$\varepsilon =c \, h^{2/3}$$ ? = c h 2 / 3 . This convergence result shows that Babuska's paradox associated to Stokes equations with slip boundary conditions is circumvented. A numerical example illustrates the theoretical results, notably that regularized normal approximations give better approximations and convergence orders.